: The Hamiltonian for the one-dimensional simple harmonic oscillator is: mw? 1 ÎĤ =- + 2m Use the definition of the simple harmonic oscillator lowering operator 1 -î + iv mwh mw V2 and its Hermitian conjugate to: (a) Evaluate (â', â] (b) Show that = Vi e n = Vmwh () and Ĥ = ħw(âtâ + }) à ât %3D (c) Evaluate [âtâ, â] (d) Find (î),(p), (r2) and (p2) for the nth stationary state of the harmonic oscillator. Check that the uncertainty principle is satisfied.
: The Hamiltonian for the one-dimensional simple harmonic oscillator is: mw? 1 ÎĤ =- + 2m Use the definition of the simple harmonic oscillator lowering operator 1 -î + iv mwh mw V2 and its Hermitian conjugate to: (a) Evaluate (â', â] (b) Show that = Vi e n = Vmwh () and Ĥ = ħw(âtâ + }) à ât %3D (c) Evaluate [âtâ, â] (d) Find (î),(p), (r2) and (p2) for the nth stationary state of the harmonic oscillator. Check that the uncertainty principle is satisfied.
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![: The Hamiltonian for the one-dimensional simple harmonic oscillator is:
mw?
1
ÎĤ =- +
2m
Use the definition of the simple harmonic oscillator lowering operator
1
-î + iv
mwh
mw
V2
and its Hermitian conjugate to:
(a) Evaluate (â', â]
(b) Show that = Vi e n = Vmwh () and Ĥ = ħw(âtâ + })
à ât
%3D
(c) Evaluate [âtâ, â]
(d) Find (î),(f), (x²) and (p2) for the nth stationary state of the harmonic oscillator. Check that
the uncertainty principle is satisfied.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff07ca1cd-42a8-4141-aaaa-b45f391f59e0%2F299274de-b712-4dec-8f5e-41ef66e33086%2F0ynkkqd_processed.png&w=3840&q=75)
Transcribed Image Text:: The Hamiltonian for the one-dimensional simple harmonic oscillator is:
mw?
1
ÎĤ =- +
2m
Use the definition of the simple harmonic oscillator lowering operator
1
-î + iv
mwh
mw
V2
and its Hermitian conjugate to:
(a) Evaluate (â', â]
(b) Show that = Vi e n = Vmwh () and Ĥ = ħw(âtâ + })
à ât
%3D
(c) Evaluate [âtâ, â]
(d) Find (î),(f), (x²) and (p2) for the nth stationary state of the harmonic oscillator. Check that
the uncertainty principle is satisfied.
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